written 6.1 years ago by | • modified 2.2 years ago |
Find the values of:
(i) f(3)
(ii) fâ(1-i)
(iii) fââ(1-i)
Subject: Applied Mathematics 4
Topic: Complex Integration
Difficulty: Medium
written 6.1 years ago by | • modified 2.2 years ago |
Find the values of:
(i) f(3)
(ii) fâ(1-i)
(iii) fââ(1-i)
Subject: Applied Mathematics 4
Topic: Complex Integration
Difficulty: Medium
written 5.8 years ago by |
Here, r = 2
$f(\xi) = 2 \pi i[3 \xi^2 + 2 \xi + 1]$ if $\xi$ lies inside
$f(\xi) = 0$ if $\xi$ lies outside
(i) z = 3 = (3,0) lies outside radius = 2. Therefore, f(3) = 0
(ii) z = (1-i) = (1,-1)
Let A = (1,-1) and c = (0,0) where c is the centre of circle $ x^2 + y^2 = 4 $
d(Ac) = $ \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1+1} = \sqrt{2} \lt 2 $
Therefore, A lies inside the circle.
$ \therefore f'(\xi) = 2 \pi i (6 \xi + 2) \\ f'(1-i) = 2 \pi i (6 (1-i) + 2) \\ = 2 \pi i (6-6i + 2) \\ = 2 \pi i (8-6i) $
(iii) z =(1-i) lies inside
$ f''(\xi) = 2 \pi i(6) = 12 \pi i \\ f''(1-i) = 12 \pi i $